A CW complex \(X\) is built by induction on dimension: the \(0\)-skeleton \(X^0\) is a discrete space, and \(X^n\) is obtained from \(X^{n-1}\) by attaching \(n\)-cells along maps \(S^{n-1}\to X^{n-1}\), the adjunction construction of constructions of spaces. The space \(X=\bigcup_nX^n\) has the weak topology: a subset is closed if and only if its intersection with every \(X^n\) is closed. Cones, suspensions, wedge sums, mapping cylinders, and mapping cones of CW complexes and cellular maps are CW complexes. The cellular approximation theorem: every map between CW complexes is homotopic to a cellular map, one sending \(X^n\) into \(Y^n\) for every \(n\). If \(X\) or \(Y\) is locally compact, \(X\times Y\) is a CW complex whose \(n\)-cells are the products \(e^i\times e^j\) of cells with \(i+j=n\).
CW complex (Tag D-ZOU5G)
A CW complex is a topological space \(X\) together with subspaces \(X^0\subseteq X^1\subseteq\cdots\) constructed as follows.
\(X^0\) is a discrete space; its points are the \(0\)-cells.
For \(n\geq 1\), the \(n\)-skeleton \(X^n\) is obtained from \(X^{n-1}\) by attaching a family of \(n\)-cells \(e^n_\alpha\) along continuous maps \(\varphi_\alpha\colon S^{n-1}\to X^{n-1}\): \[ X^n = \qty{X^{n-1} \disjoint \Disjoint_\alpha D^n_\alpha} / \qty{x \sim \varphi_\alpha(x) \text{ for } x \in \del D^n_\alpha}, \] with the quotient topology.
Either \(X = X^n\) for some \(n\), or \(X = \Union_{n\geq 0} X^n\) with the weak topology: a subset \(A\subseteq X\) is closed if and only if \(A \intersect X^n\) is closed in \(X^n\) for every \(n\).
[@Hat02].
Attaching an $n$-cell (Tag D-MMDM3)
Let \(X\) be a topological space and \(n\geq 0\). Let \(D^n\subseteq\RR^n\) be the closed unit ball and \(S^{n-1}=\del D^n\) its boundary sphere, with \(S^{-1}=\emptyset\). For a continuous map \(f\colon S^{n-1}\to X\), the space obtained by attaching an \(n\)-cell to \(X\) along \(f\) is the quotient space \[ X\cup_f D^n \coloneqq \qty{X\disjoint D^n}/\qty{s\sim f(s) \text{ for } s\in S^{n-1}}, \] the case \(X_0=X\), \(X_1=D^n\), \(A=S^{n-1}\) of attaching a space along a map. The map \(f\) is the attaching map. The composite \(\Phi\colon D^n\to X\disjoint D^n\to X\cup_f D^n\) of the inclusion and the quotient map is the characteristic map, its image \(\Phi(D^n)\) is the closed \(n\)-cell, and \(e^n\coloneqq\Phi(D^n\setminus S^{n-1})\) is the open \(n\)-cell.
In low dimensions:
\(n=0\): \(D^0\) is a point and \(S^{-1}=\emptyset\), so \(X\cup_f D^0=X\disjoint\ts{\pt}\).
\(n=1\): \(D^1=[-1,1]\subseteq\RR\) and \(f\) is a map \(S^0=\ts{-1,1}\to X\); the closed \(1\)-cell is the image of a path from \(f(-1)\) to \(f(1)\).
\(n=2\): \(D^2\subseteq\RR^2\) is the closed disk, attached along a map \(S^1\to X\).
\(n=3\): \(D^3\subseteq\RR^3\) is the closed ball, attached along a map \(S^2\to X\).
Cells of a CW complex (Tag D-MEPE3)
Let \(X\) be a CW complex whose \(n\)-skeleton \(X^n\) is obtained from \(X^{n-1}\) by attaching \(n\)-disks \(D^n_\alpha\) along attaching maps \(\varphi_\alpha\colon S^{n-1}\to X^{n-1}\). The characteristic map of \(D^n_\alpha\) is the composite \(\Phi_\alpha\colon D^n_\alpha\hookrightarrow X^{n-1}\sqcup\coprod_\beta D^n_\beta\to X^n\hookrightarrow X\), whose restriction to \(S^{n-1}\) is \(\varphi_\alpha\). The \(n\)-cell \(e^n_\alpha\) is the image \(\Phi_\alpha(D^n\sm S^{n-1})\).
Each \(n\)-cell \(e^n_\alpha\) is homeomorphic to the open disk \(D^n\sm S^{n-1}\cong\RR^n\), and the underlying set of \(X\) is the disjoint union of the cells \(e^n_\alpha\) over all \(n\) and \(\alpha\).
See [@Hat02].
Weak topology determined by a family of subspaces (Tag D-2KICH)
Let \(X\) be a topological space and let \((X_\alpha)_{\alpha}\) be a family of subspaces with \(X = \Union_\alpha X_\alpha\). The weak topology determined by \((X_\alpha)_\alpha\) is the topology on \(X\) in which a subset \(A\subseteq X\) is closed if and only if \(A\intersect X_\alpha\) is closed in \(X_\alpha\) for every \(\alpha\).
The weak topology is the finest topology on the set \(X\) for which every inclusion \(X_\alpha\injects X\) is continuous. A CW complex \(X\) carries the weak topology determined by its skeleta \(X^n\), equivalently by its closed cells.
[@Hat02].
Cellular map (Tag D-3IWX2)
Let \(X\) and \(Y\) be CW complexes with \(n\)-skeleta \(X^n\) and \(Y^n\). A continuous map \(f\colon X\to Y\) is cellular if \(f(X^n) \subseteq Y^n\) for every \(n\geq 0\).
Simplicial complex (Tag D-NJ2Y6)
Let \(N\geq0\) and let \(v_0,\ldots,v_k\in\RR^N\) be affinely independent, meaning that \(v_1-v_0,\ldots,v_k-v_0\) are linearly independent. The \(k\)-simplex \([v_0,\ldots,v_k]\subseteq\RR^N\) is the convex hull of \(v_0,\ldots,v_k\), and these points are its vertices. A face of \([v_0,\ldots,v_k]\) is a simplex spanned by a nonempty subset of \(\ts{v_0,\ldots,v_k}\); for \(0\leq i\leq k\) and \(k\geq1\), the \(i\)-th facet is \[ \del_i[v_0,\ldots,v_k]\coloneqq[v_0,\ldots,\hat v_i,\ldots,v_k]. \] A simplicial complex in \(\RR^N\) is a set \(K\) of simplices in \(\RR^N\) such that:
every face of a simplex in \(K\) belongs to \(K\);
for \(\sigma,\tau\in K\), the intersection \(\sigma\cap\tau\) is either empty or a face of both \(\sigma\) and \(\tau\);
every point of \(\RR^N\) has a neighborhood that meets only finitely many simplices of \(K\).
Simplicial map (Tag D-RU2GC)
Let \(K\) and \(L\) be simplicial complexes with vertex sets \(V(K)\) and \(V(L)\). A simplicial map \(f\colon K\to L\) is a map \(f\colon V(K)\to V(L)\) such that whenever \(v_0,\ldots,v_k\) are the vertices of a simplex of \(K\), the points \(f(v_0),\ldots,f(v_k)\), not necessarily distinct, are the vertices of a simplex of \(L\).
A simplicial map extends to a continuous map \(\abs{K}\to\abs{L}\) of underlying spaces that is affine on each simplex: \(\sum_it_iv_i\mapsto\sum_it_if(v_i)\) for \(t_i\geq0\) with \(\sum_it_i=1\) [@Mun84].
Wedge sum (Tag D-IGUUS)
Let \((X_\alpha, x_\alpha)_{\alpha\in A}\) be a family of based topological spaces. The wedge sum \(\bigvee_{\alpha\in A} X_\alpha\) is the quotient space of the disjoint union \(\coprod_{\alpha\in A} X_\alpha\) obtained by identifying all the basepoints \(x_\alpha\) to a single point, which is its basepoint. For two based spaces, \(X\vee Y\) is the quotient of \(X\sqcup Y\) identifying \(x_0\) with \(y_0\).
Let \(X\) be a CW complex with \(n\)-skeleton \(X^n\). Then \(X^n/X^{n-1}\) is homeomorphic to \(\bigvee_\alpha S^n_\alpha\), a wedge of \(n\)-spheres with one sphere for each $n$-cell of \(X\).
If each \((X_\alpha, x_\alpha)\) is a good pair, then the inclusions \(X_\alpha\hookrightarrow\bigvee_\alpha X_\alpha\) induce an isomorphism \[ \bigoplus_\alpha\tilde H_n(X_\alpha)\to\tilde H_n\qty{\bigvee_\alpha X_\alpha} \] for every \(n\).
See [@Hat02].
Cone on a space (Tag D-II4M4)
Let \(X\) be a topological space and \(I = [0,1]\). The cone on \(X\) is the quotient space \[ CX\coloneqq(X\times I)/(X\times\ts{0}), \] in which \(X\times\ts{0}\) is collapsed to a point.
If \(X\) is nonempty, then \(CX\) is contractible, and \(x\mapsto[(x, 1)]\) is an embedding of \(X\) into \(CX\).
The cone \(CS^1\) on the circle is homeomorphic to the closed disk \(D^2\), by \([(z, t)]\mapsto tz\) for \(z\in S^1\subseteq\CC\) and \(t\in I\).
See [@Hat02].
Suspension and reduced suspension (Tag D-RZKQD)
Let \(X\) be a topological space and \(I=[0,1]\). The suspension \(SX\) is the quotient of \(X\times I\) obtained by collapsing \(X\times\ts{0}\) to one point and \(X\times\ts{1}\) to another point [@Hat02]. For a based space \((X,x_0)\), the reduced suspension is \[ \Sigma X\coloneqq\frac{X\times I}{(X\times\ts{0})\cup(X\times\ts{1})\cup(\ts{x_0}\times I)}, \] the quotient of \(SX\) obtained by collapsing the segment \(\ts{x_0}\times I\) to a point [@Hat02].
\(SX\) is the union of two copies of the cone \(CX=(X\times I)/(X\times\ts{0})\) glued along their copies of \(X\times\ts{1}\cong X\).
Smash product (Tag D-R4ZCL)
Let \((X,x_0)\) and \((Y,y_0)\) be based spaces. Identify the wedge sum \(X\vee Y\) with the subspace \(X\times\ts{y_0}\cup\ts{x_0}\times Y\) of \(X\times Y\). The smash product is the quotient space \[ X\wedge Y\coloneqq(X\times Y)/(X\vee Y), \] based at the image of \(X\vee Y\) [@Hat02].
For \(m,n\geq0\) there is a homeomorphism \(S^m\wedge S^n\cong S^{m+n}\), and for a based space \(X\) the reduced suspension satisfies \(\Sigma X\cong S^1\wedge X\) [@Hat02].
Mapping cylinder (Tag D-RMQ7W)
Let \(f\colon X\to Y\) be a continuous map and \(I=[0,1]\). The mapping cylinder of \(f\) is the quotient space \[ M_f\coloneqq\qty{(X\times I)\disjoint Y}/\qty{(x,1)\sim f(x)\text{ for } x\in X} \] [@Hat02].
Let \(\iota\colon X\to M_f\) be the map \(x\mapsto[(x,0)]\) and \(r\colon M_f\to Y\) the map with \(r([(x,t)])=f(x)\) and \(r([y])=y\). Then \(\iota\) is an embedding, \(f=r\circ\iota\), and \(M_f\) deformation retracts onto \(Y\) by sliding each segment \(\ts{x}\times I\) to its endpoint \(f(x)\); in particular \(r\) is a homotopy equivalence. Thus every continuous map factors as an embedding followed by a homotopy equivalence [@Hat02].
Mapping cone (Tag D-TFVPD)
Let \(f\colon X\to Y\) be a continuous map with mapping cylinder \(M_f=\qty{(X\times I)\disjoint Y}/\qty{(x,1)\sim f(x)}\). The mapping cone of \(f\) is \[ C_f\coloneqq M_f/(X\times\ts{0})=\qty{CX\disjoint Y}/\qty{(x,1)\sim f(x)\text{ for }x\in X}, \] where \(CX=(X\times I)/(X\times\ts{0})\) is the cone on \(X\); that is, \(C_f\) is \(Y\) with \(CX\) attached along \(f\) [@Hat02].
If \((X,A)\) is a CW pair and \(\iota\colon A\injects X\) is the inclusion, then the mapping cone \(C_\iota=X\cup CA\) is homotopy equivalent to \(X/A\) [@Hat02].
Homotopy extension property (Tag D-KVAI3)
Let \(X\) be a topological space, \(A\subseteq X\) a subspace, and \(I = [0,1]\). The pair \((X, A)\) has the homotopy extension property if for every space \(Y\), every continuous map \(f_0\colon X\to Y\), and every homotopy \(g_t\colon A\to Y\) with \(g_0 = f_0|_A\), there is a homotopy \(f_t\colon X\to Y\) starting at \(f_0\) with \(f_t|_A = g_t\) for all \(t\in I\). Equivalently, every pair of continuous maps \(X\times\ts{0}\to Y\) and \(A\times I\to Y\) that agree on \(A\times\ts{0}\) extends to a continuous map \(X\times I\to Y\).
If \(A\) is closed in \(X\), then \((X, A)\) has the homotopy extension property if and only if \(X\times\ts{0}\cup A\times I\) is a retract of \(X\times I\).
Every CW pair \((X, A)\) has the homotopy extension property.
See [@Hat02].
Cellular approximation theorem (Tag T-I2X3M)
Every map \(f\colon X\to Y\) of CW complexes is homotopic to a cellular map. If \(f\) is already cellular on a subcomplex \(A\subseteq X\), the homotopy may be taken to be stationary on \(A\) [@Hat02].
Product CW structure (Tag PR-QJZDT)
Let \(X\) and \(Y\) be CW complexes such that \(X\cross Y\) with the product topology is a CW complex with cells \(e^i_\alpha\cross e^j_\beta\); this holds if \(X\) or \(Y\) is locally compact, or if both have countably many cells [@Hat02]. Then the cellular chain complexes satisfy \[ C_n(X \cross Y) \cong \bigoplus_{i+j=n} C_i(X) \tensor_\ZZ C_j(Y), \qquad e^i\cross e^j\mapsto e^i\tensor e^j ,\] with boundary \[ d(e^i\cross e^j) = de^i\cross e^j + (-1)^{i} e^i\cross de^j \] [@Hat02]. In particular, if \(X\) and \(Y\) have finitely many cells in each dimension and \(p_X(t) = \sum_n a_n t^n\), \(p_Y(t)\) count their \(n\)-cells, then \(p_{X\cross Y}(t) = p_X(t)\,p_Y(t)\).
Skeleta and homotopy groups (Tag PR-FG4DK)
Let \(X\) be a CW complex and \(k\geq 1\). The inclusion \(X^{k+1}\injects X\) of the \((k+1)\)-skeleton induces an isomorphism \(\pi_k(X^{k+1})\cong\pi_k(X)\) [@Hat02]. In particular, \(\pi_1(X)\) depends only on the \(2\)-skeleton [@Hat02], and attaching cells of dimension \(k+2\) or higher does not change \(\pi_k\).